The ratio of the distance from the center of gravity to the vertex to the distance from the center of gravity to the midpoint of the opposite side is 2: 1. The areas of the three triangles formed by the center of gravity and the three vertices of the triangle are equal. The sum of squares of the distances from the center of gravity to the three vertices of the triangle is the smallest. In the plane rectangular coordinate system, the coordinate of the center of gravity is the arithmetic average of the vertex coordinates. The center of gravity is the point where the product of the distances of three sides in a triangle is the largest.
Nature of the center of gravity:
1, and the ratio of the distance from the center of gravity to the vertex to the distance from the center of gravity to the midpoint of the opposite side is 2: 1.
2. The areas of the three triangles formed by the center of gravity and the three vertices of the triangle are equal.
3. The sum of squares of the distances from the center of gravity to the three vertices of the triangle is the smallest.
4. In the plane rectangular coordinate system, the coordinate of the center of gravity is the arithmetic average of the vertex coordinates, that is, its coordinate is ((x 1+x2+x3)/3, (y1+y2+y3)/3); Spatial rectangular coordinate system-abscissa: (X 1+X2+X3)/3 ordinate: (Y 1+Y2+Y3)/3 ordinate: (Z 1+Z2+Z3)/3.
5. The center of gravity is the point where the product of the distances from the triangle to the three sides is the largest.
6. (Leibniz formula) If the center of gravity of triangle ABC is G and point P is any point inside it, then 3pg2 = (AP 2+BP 2+CP 2)-1/3 (AB 2+BC 2+CA 2).
7. In the triangle ABC, if the straight line passing through the center of gravity G intersects with the straight lines where AB and AC are located at P and Q respectively, AB/AP+AC/AQ=3.
8. Tangents are made from three vertices of triangle ABC to a circle with the opposite side as the diameter, and the six tangents are Pi, so Pi is all on a circle with the center of gravity G as the center and R =118 (AB 2+BC 2+CA 2) as the radius.